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If a, b and c are distinct positive numb...

If a, b and c are distinct positive numbers, then the expression `(a + b - c)(b+ c- a)(c+ a -b)- abc` is:

A

positive

B

negative

C

non-positive

D

non-negative

Text Solution

Verified by Experts

The correct Answer is:
B
since, `AM ge GM`
`:. ((b +c -a) + (c +a -b))/(2) gt (b +c -a) (c +a -b)^(1//2)`
`rArr c gt [(b + c -a) (c +a -b)]^(1//2)` ....(i)
Similarly `b gt [(a + b - c) (b + c -a)]^(1//2)`...(ii)
and `a gt [(a +b -c) (c +a -b)]^(1//2)`...(iii)
On multiplying Eqs. (i), (ii) and (iii), we get
`abc gt (a +b -c) (b + c -a) (c +a -b)`
Hence, `(a + b -c) (b +c -a) (c + a - b) - abc lt 0`
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